Characterization of the normal distribution through the power of a one-way ANOVA

One-way ANOVA model with equal number of observations in each group is considered: Y-ij = mu + alpha(i) + epsilon(ij), i = 1,...,I, j = 1,...,J, where epsilon ij are independent identically distributed random variables, and alpha(1),...,alpha(I) are constrained by alpha(1) + ... + alpha(I) = 0. Assume that I greater than or equal to 3. Consider the test rejecting the null hypothesis alpha(1) = ... = alpha(I) = 0 based on the statistic ((Y) over bar(1.) - (Y) over bar(..))(2) +...+ ((Y) over bar(I.) - (Y) over bar(..))(2). It is proved that its power depends only on the noncentrality parameter alpha(1)(2) +...+ alpha(I)(2) if and only if X-ij are normally distributed. (C) 1999 Elsevier Science B.V. All rights reserved.